hw2-s.htm

Homework 2

Irrelevant Instruments

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(Due 2/12) Consider the following regression

, where $x_{t}=1,$ $\beta$ is an unknown parameter and $e_{t\text{ }}$ are . It is proposed to estimate $\beta$ by instrumental variable (IV) methods using a $l\times 1$ vector of instruments $x_{t}$ generated by

Let be the IV estimator of $\beta$ obtained with the instruments $x_{t}.$
1. Show that $z_{t}$ satisfies
  
  and
  
  and interpret them.
  
  Easy.
2. Find the limit distribution of the vector
  
  and prove your result.
  
  Under the assumptions about $e_{t}$ and $z_{t},$ this vector will be asymptotically normal. Then by a CLT (say for a martingale difference sequence, MDS)
  
  and
  
  Finally, consider the joint distribution of two elements in the $2\times 1$ vector. Any linear combination of these elements takes the form
  
  Then is also a MDS with positive variance given by
  
  satisfying
  
  for and
  
  Thus any linear combination of the two elements in the vector is asymptotically normal, implying a limiting bivariate normal distribution
3. Find the asymptotic behavior of as
  
  Recall
  
  in a vector form and is a $T\times 1$ vector of ones. Therefore the IV (GMM) is
  
  or
  
  where $W_{T}$ is a $l\times l$ weight matrix. It is clearly that , asymptotically, the ratio of quadratic forms in the two jointly normal random variables. Suppose the model is just-identified, , then
  
  It follows that by a CMT
  
  which is a Cauchy random variable since the ratio of two normals is Cauchy. Suppose . We consider the 2SLS:
  
  Thus
  
  and
  
  where
  
  It is easy to show
  
  where
  
  Therefore
4. Assume $\Sigma _{Z}=I_{l}.$ What happens to the asymptotic distribution of when That is, let after you have passed
  
  From (2) we know that
  
  as Note
  
  since
  
  This means
  
  which is only if
  
  The implication of the result is that
  
  if and then sequentially.

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